From Chapter 20 from Bodie, Kane, and Marcus 3rd edition)
(1st 2 pages were done by hand) :Know drawings of payoff diagrams
Intrinsic value = value if exercised today
value also made up of volatility and time value
Be able to follow a WSJ index listing and see that
+ - + + + -
call value =f(S,X,σ,T,Rf,div) Table 20.1 p 651
Restrictions
on Option value
Upper bound (price no one would ever pay more than this for the option)
stock price (assuming limited liability)
Lower bound = “adjusted intrinsic value”
= Stock price - PV(x) - PV(div)
= So - PV(x) - PV(div)
Logic:
Consider 2 portfolios:
1. a call with strike price X
2. a share of stock and borrowing the PV(x) as well as the PV(div)
At expiration:
|
State of |
if S<x |
if s>x |
|
value of call |
0 |
s-x |
|
|
|
|
|
Value of share |
S |
S |
|
repayment of loan |
-X-D (and x>s) |
-X-D |
|
total |
S-x-d |
S-x-d |
Option payoff is greater than leveraged stock position
therefore the price of the call must be greater stock -PV(x)-PV(div)
Early Exercise of American puts
A call option holder can close out his position by either selling the call or by exercising the option
Assume the call is in the money
You exercise and get S-X, but we just showed that you can sell it (lower bound) for S-PV(x)-PV(d)
Since PV(x)<X, you would rather sell it than exercise it.
Thus for a NONDIVIDEND PAYING STOCK A European option and an American option are virtually identical. (at least no dividends over the life of the option)
Early exercise of American Put options is possible in the case of bankruptcy. Stock falls to zero. Now you want the money now as it can not fall past zero and waiting lowers PV.
What about for a few cents? Is it worth waiting? Depends on interest rates.
Pricing Models:
A. Binomial Option Pricing Model
assume the stock can only be at one of two prices at expiration
Current stock price =$100..stock can be either $200 or $50 at the end of period.
Call with a strike of $125, T=1 year
so the payoff from this call is either 0 or $75.
Alternatively you could buy the stock and Borrow $46.30 (PV($50) if i=8%)
At year end value of stock 50 200
repay loan -50 -50
total 0 150
this is exactly double the call....we know the outlay needed to initiate this position: $100-$46.30=$53.70.
so since this is twice a call, the call price must be $26.85 If not there is an arbitrage position available.
The keys to this are the idea of a replicating portfolio and that you can perfectly hedge your position
write 2 calls....
stock value 50 200
-obligation from shorting calls 0 -2(75)
total 50 50
thus this is RF...do it should cost pv(50) = $46.30
thus 100-2c=$46.30
B. Black Scholes Pricing model
p. 662
Very widely used model!
Succinct.
Need to know partial derivatives!
The key is to understanding B-S is the N(d) term.....
risk adjusted probability of expiring in the money
most widely used model
assuming no dividends until option date
i-rates and volatility are known and any changes is predictable
stock price movement is continuous
implied volatility-backing out projected volatility
much time is spent on trying to calculate variance(forecasted) grach-egarch, agrach models.
Dealing with Dividends
two main ways
1. Assume early exercise at dividend date
2. Use ex-dividend stock price
take the higher of the two for a pseudo-American call option
Hedge ratios”
Delta
p 671
Portfolio insurance
hedge ratio constantly changes...
selling a portion of your stock and investing in T-bills. Get the same payoff as a protective put.
N(d1)= probability of ending up in the money.
also the delta hedge ratio
usually between 0 and 1.
Option positions
spreads
money spread
Time spread
Strangles
Using (and seeing) options that are not “so labeled” is critical.
Example: levered equity is a call.
Jr. Debt is similar to a bull spread.
Using B-S allows us to predict how participants will fight over things etc.
Futures:
similar to forward, but more standardized, liquid etc
Developed largely as agricultural product, now (since 1975, financial as well)
Futures exchange clearinghouse
· Clearinghouse acts as counterparty to all trades
· acts as a guarantor, oversees delivery, bookkeeper, and settlement treasurer
Profits:
Long position
Spot price at maturity-original futures price
Short position
Original futures price-spot price at maturity
Trading takes place in the “pits” (see Trading Places)
To get out of a contract: reversing trade
Maintain margin accounts. Maintenance margin is usually 75% of the initial margin
Futures are marked to market daily.
Basis = spot (t)-Future price (t,T)
where (t,T) is the future price of a contract at time t that matures at T
A buy contract specifies that the individual will accept delivery and hence “buy” the commodity. A sell contract specifies that the individual will make delivery and hence “sell” the commodity.
The units of trading vary with each commodity. For example, if the investor buys a contract for corn, the unit of trading is 5,000 bushels. If the investor buys a contract for eggs, then the unit of trading is 22,500 dozen.
Selected Commodities,
Their Markets, and Their Units of Trading
Commodity Market Unit
of Contract
Corn Chicago Board of Trade 5,000 bushels
Soybeans
Barley
Cattle
Coffee New York Coffee and Sugar Exchange 37,500 pounds
Copper Commodity
Exchange, Inc., of
Platinum New York Mercantile Exchange 50 troy ounces
Silver Commodity
Exchange, Inc., of
Lumber Chicago Mercantile Exchange 100,000 board feet
Cotton New York Cotton Exchange 50,000 pounds
The commodity exchanges are subject to regulation. Federal laws pertaining to commodity exchanges and commodity transaction laws are enforced by the Commodity Exchange Authority, which is a division of the Department of Agriculture.
Commodities are paid for on delivery. Thus, a contract for future delivery means that the goods do not have to be paid for when the individual enters the contract. Instead, the investor (either a buyer or a seller) provides an amount of money, which is called a margin (good faith deposit). The margin should not be confused with the margin that is used in the purchase of stocks and bonds.
In the commodity markets the amount of margin does not vary with the dollar value of the transaction. Each contract has a fixed minimum margin requirement. At least according to Mayo (investments)
Note that the actual numbers will not be on the test, but if you are looking for more recent numbers for your own personal use they can be found at http://WWW.FADC.COM/pdf/99specs.pdf combined with http://WWW.FADC.COM/trf_mar.htm.
Margin Requirements
for Selected Commodity Contracts at the time of publication (1994)
Margin Financial Margin
Commodity Requirement Futures Requirement
Broilers $ 500 S&P 500 $20,000
Cocoa 1,000 NYSE Composite Index 7,000
Cotton 1,000 Value Line Index 20,000
Hogs 800 Treasury Bonds 5,000
Lumber 1,200 Treasury Bills 1,500
Potatoes 500 Municipal Bonds 4,000
Soybeans 1,500
Wheat 1,000
* Small amount of margin is one reason why a commodity contract offers so much potential leverage.
* margins can change with market conditions. As volatility increase, the margins are increased
* Example
* a contract to buy wheat at $3.50 per bushel
* controls 5,000 bushels of wheat worth a total of $17,500 (5,000 X $3.50)
* must remit $1,000
* an increase of only $0.20 per bushel produces an increase of $1,000
* The percentage return on the investment is 100%.
* An increase of less than 6 percent in the price of wheat produced a return of 100 percent.
* Leverage, of course, works both ways. If the price of the wheat declines by $0.10, the contract will be worth $17,000. A decline of only 2.9 percent in the price reduces the investor’s margin from $1,000 to $500. To maintain the position, the investor must deposit additional margin. Failure to meet the margin call will result in the broker’s closing the position.
There are two margin requirements. The first is the minimum initial deposit, and the second is the maintenance margin. For example, the margin requirement for wheat is $1,000 and the maintenance margin is $750. The maintenance margin is usually 75% of the initial margin.
The margin adjustments occur daily. After the market closes, the value of each account is totaled. In the jargon of futures trading, each account is marked to the market.
Limits are imposed by the markets on the amount of price change permitted each day.
Taxes and Futures
All positions in futures are considered to have been closed at the end of the tax year. Open positions then must be marked to the market on the last day of the tax year and any paper profits taxed as if they were realized capital gains.
The profits are arbitrarily apportioned as 60 percent long-term capital gains and 40 percent short-term capital gains.
Programmed trading refers to the coordinated purchase or sales of an entire portfolio of securities.
Arbitrage refers to the simultaneous establishment of long and short positions to take advantage of price differentials.
Index arbitrage is no different, except the arbitrageur is buying or selling index futures and securities instead of pounds. If prices deviate in different markets, an opportunity for arbitrage is created.
Programmed trading index arbitrage - if stock index futures prices rise, the arbitrageurs will short the futures and buy the stocks in the index; if futures prices decline, the arbitrageurs do the opposite.
Three potential problems:
* transactions costs
* there is an obvious problem with buying or shorting all the securities in a broad-based
index. To get around this program, the arbitrageurs have developed smaller
portfolios called baskets that mirror the larger index.
· for arbitrage to be riskless, both positions must be made simultaneously
·